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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Varieties of tori and Cartan subalgebras of restricted Lie algebras
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by Rolf Farnsteiner PDF
Trans. Amer. Math. Soc. 356 (2004), 4181-4236 Request permission

Abstract:

This paper investigates varieties of tori and Cartan subalgebras of a finite-dimensional restricted Lie algebra $(\mathfrak {g},[p])$, defined over an algebraically closed field $k$ of positive characteristic $p$. We begin by showing that schemes of tori may be used as a tool to retrieve results by A. Premet on regular Cartan subalgebras. Moreover, they give rise to principal fibre bundles, whose structure groups coincide with the Weyl groups in case $\mathfrak {g}= \operatorname {Lie}(\mathcal {G})$ is the Lie algebra of a smooth group $\mathcal {G}$. For solvable Lie algebras, varieties of tori are full affine spaces, while simple Lie algebras of classical or Cartan type cannot have varieties of this type. In the final sections the quasi-projective variety of Cartan subalgebras of minimal dimension $\textrm {rk}(\mathfrak {g})$ is shown to be irreducible of dimension $\dim _k\mathfrak {g}-\textrm {rk}(\mathfrak {g})$, with Premet’s regular Cartan subalgebras belonging to the regular locus.
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Additional Information
  • Rolf Farnsteiner
  • Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany
  • MR Author ID: 194225
  • Email: rolf@mathematik.uni-bielefeld.de
  • Received by editor(s): May 23, 2002
  • Received by editor(s) in revised form: July 19, 2003
  • Published electronically: April 16, 2004
  • Additional Notes: Supported by a Mercator Professorship of the D.F.G

  • Dedicated: Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday
  • © Copyright 2004 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 4181-4236
  • MSC (2000): Primary 17B50
  • DOI: https://doi.org/10.1090/S0002-9947-04-03476-2
  • MathSciNet review: 2058843